Pressure Jump Kinetics of Disorder to BCC Ordering in Diblock

Nov 8, 2012 - Because of the fast equilibration in a pressure jump we were able to observe the initial induction stage during which the short-range or...
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Pressure Jump Kinetics of Disorder to BCC Ordering in Diblock Copolymer Micelles in a Selective Solvent Yongsheng Liu,† Julian D. Spring,† Milos Steinhart,‡ and Rama Bansil†,* †

Department of Physics, Boston University, Boston, Massachusetts 02215, United States Institute of Macromolecular Chemistry, Academy of Sciences of the Czech Republic, Heyrovsky Sq. 2, 162 06 Prague 6, Czech Republic

‡

ABSTRACT: We present time-resolved small-angle X-ray scattering (SAXS) measurements of the kinetics of a barotropic disorder−order transition in polystyrene−polyisoprene (SI) diblock micelles in diethyl phthalate (DEP), a styrene selective solvent. Because of the fast equilibration in a pressure jump we were able to observe the initial induction stage during which the short-range order of the micellar fluid increases, followed by a sharp first-order nucleation and growth of the BCC phase and eventual late stage coarsening. The time evolution of intensity from individual diffraction spots of a crystallite was obtained from the analysis of 2-dimensional SAXS patterns and shows late stage power-law growth. The time evolution of the SAXS intensity profiles was analyzed by fitting the contribution to the scattering from the ordered Bragg peaks by Gaussians and analyzing the contribution to the scattering from the disordered micellar liquid in terms of the Percus−Yevick interacting hard-sphere model. The time evolution of the micellar structural parameters obtained from this analysis reveals that the disorder to order transition occurs when the volume fraction of micelles reaches 0.53, and the effective hard sphere radius and volume fraction of micelles increases upon ordering while the core radius is unaffected. In both the ordered and disordered states the hard sphere radius and volume fraction of micelles increase with increasing pressure indicating increased swelling of the micellar corona. In the ordered state there is considerable penetration of the corona chains from neighboring micelles.

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decomposition in binary fluids.12 The early stages of nucleation including the formation of the critical nucleus in a polymer blend were observed by small angle neutron scattering as a function of pressure.13 To the best of our knowledge there are no studies of pressure-jump kinetics in block copolymers, although a barotropic disorder → order transition following depressurization was reported in an SI diblock.8 However, the kinetics of the transition could not be examined due to the limited time resolution of the data. In this paper we present a time-resolved SAXS study of the kinetics of the disorder to order transition in a SI diblock (16K−11K) in a styrene selective solvent, diethyl phthalate (DEP), following a pressure jump (P-jump). The diblock forms spherical micelles with polyisoprene in the cores and polystyrene in the corona. We examine the initial stages of nucleation and growth from the time evolution of the SAXS data, and use the Percus−Yevick model of interacting hard spheres to obtain the pressure dependence of micellar parameters.

INTRODUCTION It has long been known that pressure jump techniques offer a unique advantage over temperature jump methods for investigating the initial stages of the kinetics of a phase transition process, because pressure changes equilibrate at the speed of sound (milliseconds) as compared to tens of seconds or minutes for a T-jump due to the finite heat capacity of the sample holder. Moreover, depressurization or increased pressurization are both equally fast, unlike heating and cooling. From a practical point of view pressure changes offer an alternative to temperature for controlling the structure and processing of block copolymer materials.1 Small angle neutron scattering and SAXS studies of thermodynamics of the order− disorder transition in block copolymer melts have shown that pressure influences the order−disorder transition temperature (ODT) as well as the chain length and degree of segregation.1−5 The ODT can increase2,4−6 or decrease3,6 with increasing pressure depending on the chemical composition. Re-entrant7−9 and inverted phase diagrams have also been observed.3 Pressure changes in solutions have been shown to affect thermal fluctuations in solutions of SI in a neutral solvent,10 and to induce aggregation11 and promote swelling of micelles in aqueous block copolymers.9 Pressure jump experiments played a key role in establishing the differences between nucleation and growth and spinodal © 2012 American Chemical Society

Received: July 24, 2012 Revised: October 8, 2012 Published: November 8, 2012 9147

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Figure 1. Sketch of pressure apparatus adapted from Steinhart et al.14 Two electronic relay gas valves, R-RGV for releasing pressure and P-RGV for pressurization, connected to a pneumatic valve are used to produce pressure jumps in the high pressure cell. The valves are controlled by a PC computer interface. The pressure generator is a large stainless steel cylinder with a piston that can be driven either by a computer controlled motor or manually with a handle attached to the piston.

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and half-width at half-maximum (w) were determined by a leastsquares fitting procedure using the “lsqcurvef it” function in Matlab program. The same program was used for determining the micellar parameters using the Percus−Yevick model of interacting hard spheres. Sample Preparation. The samples used in this study were prepared from a diblock copolymer poly(styrene-b-isoprene), (Polymer Source, Inc., molecular weight 16K−11K, and polydispersity 1.02) at a concentration of 30% (w/v) in DEP (Aldrich-Sigma) which is selective to PS blocks. To reduce both the phase transition speed and the ODT, homopolymer PS (Polysciences, molecular weight 4630) was added at a concentration of 20% in DEP. The addition of homopolymer PS increases the solvent viscosity and thus slows down the phase separation process which completed in ∼40 min, compared to ∼5 min for a similar SI sample without PS.16 The sample with added PS showed a BCC phase with the ODT lowered by about 3 °C, in agreement with earlier reports.17 Methylene chloride (dichloromethane, 99.6%) was added to the mixture as a cosolvent and gently shaken in a vortex mixer until a clear solution was obtained. Methylene chloride was removed by evaporation until there was no weight change of the sample over at least 24 h.

EXPERIMENTAL SECTION

High-Pressure Instrument. We built a pressure control apparatus (Figure 1) for SAXS measurements based on the design by Steinhart et al.14 Pressure from atmospheric up to 0.35 GPa (3.5 kbar) can be produced by a motor-driven, piston-type generator (High Pressure Inc., PA) and transferred through a network containing pressurizing liquid to a sample cell. The cell with optical path length of 1.6 mm, has two diamond windows with reasonable transmission for the 9.01 keV X-ray beam and low background scattering. Scattering can be observed at angles up to 30° from solid or liquid samples. The temperature of the cell is controlled over the range of −10 to +200 °C using Peltiers with a circulating water bath as a heat exchanger. The pressure and temperature controllers are interfaced, so that we can make temperature ramps and temperature jumps at fixed pressure, or make pressure jumps at fixed temperature. For the block copolymers in organic solvents we use water as the pressurizing fluid. The samples are isolated from the pressurizing medium by Teflon pistons. The high-pressure network consists of two sections separated by a pneumatic valve with the inner section connected permanently to the cell and the outer one to the pressure generator. For pressure jumps, the outer section is brought to a different pressure level from the inner one and the jumps are accomplished by opening the connection between both sections using electronic relay valves. Small Angle X-ray Scattering. SAXS experiments were carried out at beamline X10A of National Synchrotron Light Source (NSLS) at Brookhaven National Lab (BNL). The X-ray wavelength was 0.1089 nm (9.01 keV) with energy resolution of 1.1%. A two-dimensional (2D) CCD detector with an array of 1024 × 1024 was used to record the scattering pattern. Azimuthal integration of the image was performed to obtain the scattering intensity I(q) versus the scattering vector q = (4π/λ) sinθ, with 2θ being the scattering angle. The intensity was normalized to constant beam intensity and the contribution from solvent was subtracted. In our experimental setup with a sample-to-detector distance of 2.1 m, we cover a q range of 2 nm−1 with q = 0 close to the center of the 1024 × 1024 pixel CCD detector. The high pressure system (described below) was installed inside the hutch of the beamline, and pressure and temperature controlled from outside the hutch so as to change pressure or temperature without removing the sample from the beam. Temperature ramp SAXS measurements at a rate of 1 °C/minute were performed at fixed pressure to determine TODT, the order−disorder transition temperature. Time-resolved SAXS measurements of pressure jumps at fixed temperature were carried out to study the kinetics of disorder to BCC phase transition. The SAXS intensity was imaged on a 2-dimensional CCD camera with a time-resolution of 30 s per frame during the T-ramp, and at a faster rate of 10 s per frame for P-jump. Data acquisition and processing methods are described in previous publications.15 The peak position qmax, maximum intensity I0,

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RESULTS AND DISCUSSION Pressure Dependence of TODT. To identify the different phases we performed SAXS measurements from samples equilibrated for half an hour or longer at different values of pressure keeping the temperature fixed. Figure 2a shows typical data for SAXS intensity I(q) versus the scattering wavenumber, q over the pressure range of 100−1000 bar at a fixed temperature of 103 °C. The broad peak at 100 bar is due to disordered micelles. At 200 bar, sharp Bragg peaks superposed on a broad peak are visible. The appearance of the primary BCC [100] peak and the second BCC [200] peak indicates that a barotropic transition from disorder to BCC occurs between 100 and 200 bar. The ODT transition temperature, TODT was determined from time-resolved SAXS measurements following a temperature ramp (T-ramp) at fixed pressure. The evolution of the scattering intensity during a typical T-ramp measurement is shown in Figure 2b and clearly identifies the TODT. We note that the transition temperature measured in a T-ramp is higher than the value obtained in quasi-static measurements from samples annealed at each temperature for a long time before making the SAXS measurement. The T-ramp method is much faster and provides a reliable estimate of TODT for determining between which two states to perform a jump to follow the kinetics. From similar measurements made at different 9148

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volume at TODT via the Clausius−Clapeyron equation for a first order phase transition. Pressure Jump Kinetics. To investigate the kinetics of the disorder to order transition we performed time-resolved SAXS measurements following a pressure jump (P-jump) from 100 to 200 bar at fixed temperature of 103 °C. The sample is initially in the disordered phase because TODT at 100 bar is 102 °C, and eventually it is in the ordered state as TODT at 200 bar is 108 °C . In most studies of phase transition via nucleation and growth, an induction period is observed during which the material remains in the metastable stage. In this experiment because the pressure equilibrates in less than one second we were able to observe the initial stages of the phase transition process immediately following the pressure jump. As shown in Figure 3a, the sample remains in the disordered state initially and takes

Figure 2. Barotropic disorder−order transition. (a) SAXS intensity I(q) vs q in BCC phase at fixed temperature of 103 °C as a function of pressure. (b) Time-evolution of SAXS profile during a T-ramp from 95 to 110 °C at a rate of 1 °C/min at 100 bar. The ODT of BCC → disorder transition is determined to be 102 °C. (c) The pressure dependence of TODT and the primary [110] peak position, qmax at fixed temperature of 103 °C. Dashed lines are for viewing purposes.

Figure 3. Pressure jump kinetics of disorder to order transition. (a) Time-resolved SAXS following a pressure jump of 100 to 200 bar at 103 °C. (b) Time-evolution of the primary BCC peak intensity (Δ) and width (□) during the phase transition. (c) Time-evolution of primary BCC [110] peak position (○) during the phase transition. (d) Time-evolution of [200] BCC peak position (Δ) during the phase transition. Pressure jump was finished within 1 s (labeled by vertical dotted line). The dashed vertical lines indicate the three stages in the ordering transition, the solid line in Figure 2(b) is the fit to the Avrami model in stage II and the dashed line is the power law fit to ξ = 1/w. See text for details.

pressures we observed that TODT increases with increasing pressure as shown in Figure 2 c with ΔTODT/ΔP = 16 °C/kbar. Previous work16 on a similar SI diblock (18K−12K) at 30% in DEP mixed with homopolymer PS showed that ΔTODT/ΔP increases slightly from 17 to 20 °C/kbar as the PS content of the solvent is varied from 0 to 20%. The position of the primary peak qmax decreases with increasing pressure, implying that the lattice constant, a, increases from 29.4 to 30.4 nm over ΔP = 500 bar, i.e. ∼ 2 nm/kbar. A similar increase in TODT and decrease in primary peak position with increasing pressure have been reported in SI diblock melt in the lamellar phase5 and interpreted as arising due to the discontinuous change in

about 2.5 min for the [200] BCC peak at √2qmax to appear, indicating the formation of a BCC lattice. To describe the kinetics of the transformation we characterized the SAXS data by determining the peak intensity I, width (w), and position qmax of the primary peak. This peak represents the short-range local order in the disordered state and transforms to the [110] BCC peak in the ordered state. Figure 3b shows the time 9149

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Figure 4. Two-dimensional SAXS images at (a) 5.5 min, (b) 8.25 min, (c) 16.5 min, and (d) 55 min showing the development of individual BCC crystals following the P-jump from 100 to 200 bar. The image at 5.5 min (a) corresponding to stage II indicates diffuse scattering from a large number of randomly oriented small crystallites (see text). The bright spots visible at 8.25 min (b) indicate the appearance of a few large crystallites which grow in intensity (c, d). (e) The growth of individual spots can be monitored from a plot of the intensity versus azimuthal angle over the diffraction ring at the different times corresponding to the images in parts a−d. (f) Intensity of selected peaks (labeled by colored arrows in part e) versus time clearly shows late stage power-law growth.

Stage II. Nucleation and Growth. From 2.5 to ∼20 min the BCC peak intensity increases, at constant qmax. At about 2.5 min the second order BCC peak appears and grows in intensity. The increase of the primary BCC peak intensity in Stage II fits well with the Avrami equation that is conventionally used to describe nucleation and growth:19−21

evolution of the peak intensity and its width, while the change in peak position with time is shown in Figure 3c. Figure 3d shows the growth of the intensity of the BCC [200] peak which appears around 2.5 min and is absent in the disordered micellar fluid. As seen in these figures, the kinetics may be separated into three stages, which we interpret as: (I) the induction stage, (II) nucleation and growth of the BCC phase, and (III) rapid growth of BCC phase due to late stage coarsening. Stage I. Induction Stage. Immediately following the P-jump the micellar fluid is in a metastable state. During the initial time interval of approximately 2.5 min following the P-jump the peak position qmax increases slightly (Figure 3c) indicating that micelles move closer thereby increasing the number density, while the width w decreases quite rapidly after 1 min. There are many factors that contribute to the width of a SAXS peak. The instrumental broadening and inherent width of the beam is quite small (about 1/10 of the measured widths), and more importantly do not change during a P-jump. Thus, the decrease in width is largely due to increase in the local, i.e., short-range order in the micellar fluid, although long-range BCC order has not yet been established because higher order peaks cannot be detected. We define ξ = 1/w, as the characteristic length scale over which there is local order. We find that ξ(t) − ξ0(t = 0) increases as a power-law in time t, ξ (t) = ξ0 + b(t/τ)n with n ∼ 3.5, ξ0(t = 0) ∼ 18 nm and B = b(1/τ)n = 2.5 nm·s−n. Our observations clearly show that during the so-called induction stage the micelles move closer together reaching the optimal distance for the BCC lattice to form and short-range order increases, implying that locally nearest neighbor micelles adopt the configuration needed to nucleate the BCC state preceding the nucleation and growth of the ordered phase and the onset of long-range BCC order in agreement with the theoretical prediction of Klein18 on nucleation in near mean-field systems.

n

I(t ) − I(t0) = (I(t∞) − I(t0))(1 − e−k(t − t0) )

(1)

The data fit reasonably well to n = 2, as indicated in Figure 3b; the best fit is obtained with n = 1.6. The width w in this stage at first increases slightly and then decreases reflecting the increase in size of the polycrystalline domains. Stage III. Late-Stage Coarsening. After about 20 min we observed a very rapid increase in the intensity while qmax remains almost unchanged. The peak intensity grows like a power-law in this stage, scaling as I ∼ t1.2. This rapid increase in intensity is related to a change in the diffraction pattern as shown by the 2-dimensional diffraction images at various times in Figure 4a−d. Initially, the pattern is a diffuse ring arising from diffraction in all azimuthal directions from a large number of small randomly oriented domains which coalesce to form a few large grains giving rise to well-defined diffraction spots. The number of spots from each crystallite depends on the orientation of the BCC 110 crystal planes relative to the beam direction and the imaging plane. The intensity of the spots can be measured from a plot of the intensity versus azimuthal angle for the 110 ring, averaged over a small range of q values spanning the diffraction ring as shown in Figure 4e at a few values of time. The images in Figures 4, parts c and d, and the corresponding azimuthal plot in Figure 4e suggest that there may only be a few oriented crystallites producing the spots; however, we have not attempted to index the spots to individual crystallites since we do not know the orientation of 9150

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locally ordered regions before the onset of long-range BCC order. Percus−Yevick Analysis. As shown in Figure 2a and Figure 3a, the SAXS data show a broad peak characteristic of a disordered micellar fluid superposed with sharp Bragg peaks. This is similar to the results we had in our earlier work on disorder−order kinetics in triblocks following a temperature jump.24 We follow the same approach as used in Nie et al.24 to describe the scattering intensity during the phase transformation, I(q, t) as a sum of scattering from disordered and ordered phases, which is expressed as

the crystal planes relative to the beam and the imaging plane. The azimuthal intensity distribution for images corresponding to stage II is almost flat at 5.5 min, and by the end of stage II at 8.25 min (Figure 4b) sharp variations in intensity are discernible with a slight decrease in the baseline value, indicating a slight decrease in the diffuse scattering as a few bright spots corresponding to individual crystallites appear. Pronounced peaks and a significant depression of the baseline intensity can be noticed for the azimuthal distribution for the images c and d in Figure 4 from stage III. The time evolution of the intensity at a few selected spots indicated by arrows on Figure 4e is shown in Figure 4f. The intensity at all of the spots is very low initially. With time some spots increase in intensity while some decrease, and others show nonmonotonic changes. The intensity of two of the spots shown in Figure 4f, increases rapidly around 10 min while the third spot only appears after 30 min. The decrease in intensity could be due to rotation or movement of the crystallite relative to the beam or due to reversible addition and breaking up of small crystallites. The depression of the baseline intensity concomitant with the growth of sharp peaks from crystallite reflections confirms that the small randomly oriented domains merge to produce the large crystallites. By examining the distribution of intensity over the spot in both the radial (q) and azimuthal directions, we note that the radial profile narrows as expected for growing domains. The azimuthal intensity data for individual spots clearly shows power-law growth in stage III with exponents varying between 0.6 to 1.6 for the three different growing spots shown in Figure 4f (linear on a log−log plot). Since the characteristic linear size of an ordered domain d, is expected to grow as a power law in time, d ∼ tα with the exponent α ranging from 0.1 to 1 depending on the mechanism of growth,19,20,22,23 the total SAXS intensity from a grain of scattering volume Vs ∼ d3 should scale as I ∼ Vs2 ∼ d6 ∼ t6α. Although the azimuthal intensity data for individual spots clearly shows power-law growth, the exponent for an individual spot’s power-law growth cannot be directly equated to 6α as each spot corresponds to a specific scattering plane while the models are for overall crystallite growth. Moreover the volume Vs would be related to the projection along specific directions which could change as the crystallite moves or rotates relative to the beam. To the best of our knowledge power law growth in intensity of individual spots has not been reported in the literature. In this late stage growth the width w obtained by fitting the azimuthally averaged peak (see Figure 3) is not an accurate estimate of domain size because the scattering is dominated by the formation of a few large crystallites. Although the sharp peaks from the crystallites appear to get narrower it is difficult to accurately estimate the width of the peaks and size of domains from these large crystallites because the relative size of the crystallite and the smeared incident beam would have to be factored in determining width. A more detailed analysis of growth of individual reflections compared with modeling and simulations will be a separate study. The overall time evolution following the P-jump is similar to our previous report on kinetics of disorder to BCC phase transition in a triblock copolymer in a PS selective solvent following a temperature jump (T-jump) at atmospheric pressure.24 However, the much shorter equilibration time of the P-jump (1 s as compared to 1 min for T-jump) enables us to clearly identify the structural rearrangements that occur in the initial induction stage and observe the development of

I(q , t ) = Idisorder(q , t ) + Iorder(q , t )

(2)

The scattering from the Bragg peaks due to the ordered BCC phase is represented by a sum of Gaussians: 2

Iorder(q , t ) = ∑i AG , i (t )e−(1/2)((q − qi)

/ σi2)

(3)

with AG,i, qi, σi denoting the amplitude, position and width of the ith Gaussian peak, respectively. To describe the scattering from the disordered phase we use the Percus−Yevick (P−Y) model treating the disordered micellar fluid as a system of interacting hard spheres as has been used in several studies on block copolymers.24−29 The scattering intensity from disordered micelles can be written as a product of form factor P(q) and structure factor S(q):

I(q) = I0P(q)S(q)

(4)

Assuming the cores are spherical with radius Rc, the spherical form factor P(q) is used ⎤2 ⎛ 4π ⎞2 ⎡ 3 ⎜ − P(q) = ⎢ (sin( qR ) qR cos( qR )) Rc 3⎟ c c c ⎥ 3 ⎝ ⎠ 3 ( qR ) ⎦ ⎣ c (5)

The Percus−Yevick structure factor S(q) is used to describe the excluded-volume interaction between hard spheres, as a function of volume fraction φ and hard-sphere radius Rhs.25−29 S(q) =

1 1 + 24φG(2qR hs , φ)/(2qR hs)

(6)

where G is a trigonometric function of x = 2qRhs and φ. The hard sphere radius is a measure of the effective size of a sphere representing the entire micelle, core plus corona, and φ is the volume fraction of the effective hard spheres. A typical example of the P−Y fit is shown in Figure 5 which displays the sharp contribution of the ordered phase and the broad contribution from the disordered phase. From P−Y analysis, we obtain the core radius Rc, hard sphere radius Rhs and volume fraction φ of the micelles. We note that Rc is not as well determined from the fit as the Rhs (most likely because the minimum of the form factor cannot be clearly identified, whereas the structure factor fits very well to the observed I(q)). The error from the finite resolution of q in the experiment is about 0.8% in the vicinity of maximum peak intensity, which is comparable to the error in Rhs from P−Y fitting. Figure 6a shows the results of P−Y fitting to the P-jump data of Figure 3a. For 1 min following the quench there is hardly any change in the micellar parameters. Between 1 and 2.5 min after the P-jump, φ increases from 0.45 to 0.53, the predicted value for a BCC transition,25,30 signaling the onset of a first order transition, followed by a very rapid, sigmoidal increase in φ to the final value of 0.75. Rhs increases from 21 to 22 nm during 25−29

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Figure 5. Typical fit to the scattering data I(q) vs q with the Percus− Yevick model as described in the text. See eqs 2−6.

Figure 7. (a) Time evolution of Rhs, 2Rc, and φ obtained by Percus− Yevick fitting of SAXS data for a T-ramp from 100 to 110 °C at fixed pressure of 200 bar. (b) The temperature versus time for the T-ramp from 100 to 110 °C at at fixed pressure of 200 bar.

thermally induced order−disorder transition at atmospheric pressure.24 Although the samples do not reach equilibrium at the end of a T-ramp, the P−Y analysis of the contribution of the disordered micelles to the total scattering in the initial ordered state at 100 °C and in the final disordered state at 110 °C gives some indication of the general trends of the micellar parameters dependence on pressure at fixed temperature. The results obtained from T-ramps at different pressures (100, 200, 300, and 500 bar) are summarized in Table 1. As explained earlier Rc is not as well determined as Rhs and within fitting error does not appear to change systematically with pressure in either state. In both the ordered and disordered states Rhs and φ increase with increasing pressure. At all the pressures examined Rhs and φ increase upon ordering. In previous studies of pressure dependence in SI melt4 and SI in neutral solvent10 it has been shown that the PS chains stretch due to both a slight increase in Rg, the radius of gyration4 and the influence of pressure on the degree of segregation4,10 The changes in Rg with pressure are small (∼0.1 nm/kbar) compared to the effect of pressure on the degree of segregation. Previous work at atmospheric pressure has shown increased swelling of the corona31 and this effect is also likely to be enhanced at higher pressures. Thus, the combined effect of pressure on the block copolymer chains and their degree of segregation leads to increased swelling of micelles with increasing pressure. The increase in φ is due to the larger volume per micelle, as well as to the increase in number density of micelles due to compressibility effects. We compare the radii Rc and Rhs with the unperturbed chain dimensions calculated with PI blocks treated as Gaussian chains, LPI = aPINPI0.5, and PS blocks treated as self-avoiding random walk chains, LPS = aPSNPS0.6, where a is the monomer length, and N is the polymerization number. The monomer length is calculated from specific volume, v = (4π/3)a3, where vPS = 1.40 nm3 and vPI = 1.16 nm3, which gives aPS = 0.69 nm and aPI = 0.65 nm, respectively.31 Using NPS = 16 100/104 = 155 and NPI = 11 200/68 = 165, where 104 and 68 are the molar mass for PS and PI respectively, we obtain LPS ∼ 14 nm and LPI ∼ 8 nm. A comparison of Rc and Rhs with the unperturbed chain dimensions reveals that Rc ∼ 7.5 nm is slightly smaller than the unperturbed length of PI blocks at atmospheric pressure (∼8 nm). Treating the PS blocks in the solvent DEP as self-avoiding random walk chains we obtain, LPS

Figure 6. Time-evolution of micellar parameters in a P-jump. (a) Time evolution of Rhs, 2Rc, and φ obtained from P−Y fitting of the SAXS data for P-jump from 100 to 200 bar at fixed temperature of 103 °C. (b) Contribution of Iorder(qmax, t) and Idisorder(qmax, t) at the peak maximum obtained by fitting SAXS data for P-jump to eqs 2−6.

the transition, while Rc remains unchanged throughout the transformation. The time of 2.5 min as the onset of the BCC transition is the same as that identified from the peak shape analysis (Figure 3) showing an induction stage (I) followed by nucleation and growth (stage II). Figure 6b shows that the contribution of the disordered phase to the peak intensity, Idisorder decreases during stage I while that of the ordered phase, Iorder increases in stage II. The contribution of the disordered micelles to the total scattering intensity is very small after the onset of ordering and growth of the ordered phase. The reverse transition from order → disorder was too fast to obtain time-resolved SAXS data following a depressurization jump. Thus, we examined the nonisothermal kinetics of the order → disorder transition by T-ramp measurements at various fixed values of pressure from 100 to 500 bar. The results of a P−Y fit to T-ramp data from 100 to 110 °C, at fixed pressure of 200 bar, are shown in Figure 7a. The ODT is identified at ∼108 °C. Figure 7b shows the temperature in the T-ramp as the sample was heated at 1 °C/min until a little past the ODT after which the sample continued to disorder at constant temperature of 110 °C. The P−Y analysis shows that within fitting error Rc does not change, while φ decreases sharply at the onset of melting. Rhs also shows a discrete change at the transition. Similar results were obtained in our previous study of a 9152

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Table 1. Micellar Parameters Obtained by Fitting the SAXS Data for the Ordered and Disordered Phases at the Beginning and End of a T-Ramp Using Eqs 2−5 pressure (bar)

temperature (°C)

phase

100 200 300 500 100 200 300 500

110 110 110 110 100 100 100 100

disorder disorder disorder disorder BCC BCC BCC BCC

core radius Rc (nm) 7.6 7.8 7.4 7.8 7.4 8.0 8.0 7.8

± ± ± ± ± ± ± ±

hard sphere radius Rhs(nm)

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

21.2 21.8 22.7 23.6 22.4 22.8 23.9 25.1

± ± ± ± ± ± ± ±

0.1 0.1 0.1 0.2 0.01 0.01 0.03 0.05

hard sphere volume fraction (φ) 0.495 0.506 0.519 0.535 0.692 0.713 0.727 0.726

± ± ± ± ± ± ± ±

0.003 0.003 0.003 0.003 0.001 0.001 0.001 0.004

transition. At ∼2.5 min for a P-jump from 100 to 200 bar the appearance of the second order BCC peak provides a clear signature of nucleation and growth of the ordered phase. Analysis of the SAXS data using the Percus−Yevick model for interacting hard spheres to describe the disordered micelles, shows that the disorder to order transition occurs when the volume fraction of hard spheres reaches 0.53, and the volume fraction of micelles continues to increase as ordering proceeds. The core radius of the micelle is essentially unchanged upon ordering, while the hard sphere interaction radius Rhs increases upon ordering. The growth of the SAXS intensity during the nucleation and growth stage agrees with the Avrami model. Late stage coarsening was also observed with power law growth in the intensity and the appearance of bright diffraction spots in 2-dimensional diffraction patterns corresponding to specific reflections from large crystallites formed by coalescence of the small randomly oriented domains. The time evolution of the intensities of these spots showed nonmonotonic changes as crystallites merge and break apart. Those spots that grew exhibited a power law growth in intensity with exponents varying from 0.6 to 1.6. The reverse transition from order to disorder was too fast to follow the kinetics by time-resolved SAXS measurements. Instead we examined the nonisothermal order to disorder kinetics by making T-ramp measurements at fixed pressures. From Percus−Yevick analysis of these data we obtained the dependence of the micellar parameters on pressure. We note that in SI diblocks both the micelle size and volume fraction increase as pressure increases, suggesting that pressurization may be used for swelling micelles rapidly (or conversely depressurization for rapid shrinking). In the SI diblock system TODT increases with increasing pressure thus a disorder to order transition can be triggered by increasing the pressure and the reverse by depressurizing. The disorder to order transition occurs in a couple of minutes while the reverse transition is even faster. These time scales on the order of few minutes or less may be convenient for biomedical and materials processing applications using pressurization/depressurization to deliver micellar contents or change morphology and rheological states.

∼ 14 nm which gives the overall micelle radius ∼ LPI + LPS ∼ 22 nm, close to the f itted value of Rhs. As mentioned earlier, the lattice constant in the BCC phase, a increases from 29.4 to 30.4 nm over ΔP = 500 bar. Comparing a to 2Rhs we conclude that there is significant overlapping of the PS chains in coronas of adjacent micelles in the BCC state. The increase of Rhs with pressure (see Table 1) also implies that increasing pressure leads to increased penetration of the chains in the corona. The overlapping of corona chains is a common feature of block copolymer micelles; the present work shows that chain interpretation is enhanced by increasing pressure. The sketch in Figure 8 illustrates the effect of pressure in the trans-

Figure 8. Sketch of the effect of pressure on the micelles undergoing a disorder → order transition. The micelles are characterized as PI cores of radius Rc with a corona of PS chains. For visual clarity only a few chains are shown. The hard sphere radius Rhs, of Percus−Yevick model is indicated. Also shown are linear homopolymer PS chains in the solvent DEP. The PS chains of the two micelles at the top are shown with different colors to indicate the penetration of the corona chains between adjacent micelles. The overlap volume is larger at higher pressure indicating greater penetration.

formation from disordered to ordered micelles, showing the increase in Rhs, as well as the increased penetration of the corona PS chains between adjacent micelles upon ordering.

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AUTHOR INFORMATION

Corresponding Author

CONCLUSIONS In conclusion, we note that due to the faster equilibration time of pressure changes we were able to observe the induction stage prior to nucleation of the BCC state and follow the kinetics of the disorder-BCC transition in a micellar SI diblock solution in a selective solvent following a pressure jump by time-resolved SAXS measurements. The SAXS data show that during the induction stage a local structural rearrangement in the metastable micellar fluid precedes the onset of the ordering

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Dr. Steve Bennett for valuable assistance at x10A beamline of NSLS and acknowledge very helpful discussions with Prof. Bill Klein of Boston University. This research was 9153

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Macromolecules

Article

(29) Yurekli, K.; Krishnamoorti, R. Macromolecules 2003, 35, 4075− 4083. (30) Pusey, P. N.; van Megan, W. Nature 1986, 320, 340−342. (31) Han, C. D.; Kim, J.; Kim, J. K.; Chu, S. G. Macromolecules 1989, 22, 3443−3451.

supported by NSF Division of Materials Research award No. DMR 0804784. The SAXS measurements were carried out at NSLS of BNL, which is supported by the U.S. Department of Energy, Division of Materials Sciences and Division of Chemical Sciences under Contract No. DE-AC02-98CH10886.

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